Cfd Class April 13 2010

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Finite Volume Method in CFD software packages

About 80% of commercial CFD packages utilize FVM

Example: Steady-state specie convection:

Mass balance over the control volume contains

values at the faces. These values have to be determined

from interpolation of the values in the cell centers:

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Example- continued (discretization)

Interpolation assumption: the value at the face is equal to the value in thecenter of the cell upstream of the face: Upwind scheme .

Notation:

Face areas:

Concentration at the

centers:

Components of 2D velocity at

the centers:

Source at the center: Diffusion:

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Example- continued (algebraic equations)

Re-arrangement:

Simplified notation (index nb refers to the neighboring centers) :

or

The concentration is calculated by recalculating cP from the equation iteratively

for all cells in the domain.

How many members in the sum if mesh is tetrahedral ?

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Residuals and convergence

Depending of the interpolation scheme thecoefficients in final equation may depend on the

unknown function so equation: becomes non-linear

The equation is solved by an iterative process. Absolute residuals measure

imbalance in conservation equations at every iteration:

Scaled residual in single cell

Scaled residual of a mesh

Can quality of mesh affect convergence ?

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The residuals show how fast the solution (of discretized algebraic system)converge

Small values of the residuals are necessary but not sufficient conditions to stop

the iterations

Solutions of CFD problems are considered to converge when the flow field andscalar fields are no longer changing.

Common practice is to monitor residuals and averaged flow parameters:

Residuals and convergence

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First well-documented use was by Evans and Harlow (1957) atLos Alamos and Gentry, Martin and Daley (1966): efficient, iterative

solvers are well developed.

Stable: even if variable fields are not smooth across shocks and

other discontinuities, mass, momentum and energy are alwaysconserved.

FVM has an advantage in memory use and speed for very large

problems, higher speed flows, turbulent flows, and source term

dominated flows (like combustion).

Basic FV control volume balance does not limit cell shape; mass,

momentum, energy conserved even on coarse grids;

Disadvantages: false diffusion, i.e. initial sharp distribution of

variables are smeared out during the solution

Finite Volume Method: pros and contras

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Example: artif icial dif fusion of temperature in 1D velocity field

www.bakker.org

Artificial diffusion is caused by numerical rounding during the iterat ions

If the thermal diffusivity coefficient is

set to zero, the temperature will be

exactly 100 C everywhere above the

diagonal and exactly 0 C everywherebelow the diagonal.

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Example: arti ficial diffusion: temperature contour plot

Grid refinement coupled with a higher-order interpolation scheme will

minimize the false diffusion.

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The value at the face is the same as

the cell centered value in the cell

upstream of the face.

The main advantages are that it iseasy to implement and that it results in

very stable calculations, but it also

very diffusive. Gradients in the flow

field tend to be smeared out

This is often the best scheme to start

calculations with.

First-order upwind scheme

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Central Differencing Scheme

The value of unknown function is

determined at the face by linear

interpolation between the cell centered

values.

The scheme often leads to

oscillations in the solution or

divergence if advection dominates

diffusive transport:

Switching to first order upwind in

cells where >2

is called a hybrid scheme.

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The value of the function is

interpolated from the cell values in the

two cells upstream of the face.

In regions with strong gradients it can

result in face values that are outside ofthe range of cell values. It is then

necessary to apply limiters to the

predicted face values.

Second-order upwind with limiters isone of the most popular numerical

schemes because of its combination of

accuracy and stability.

Second-order upwind scheme

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QUICK: A quadratic curve is fitted through two upstream nodes andone downstream node.

Power Law Scheme: The face value is determined from an

exponential profile through the cell values.

Different pressure-velocity (SIMPLE, SIMPLEC, PISO) coupling

algorithms are used to derive equations for the pressure from the

momentum equations and the continuity equation.

It is recommended to start calculations with first-order upwind and

after about 100 iterations to switch over to second-order upwind.

Other schemes may speed up convergence but instability is

possible

Other schemes and solution tactic (ANSYS FLUENT)

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Suppression of osci llation in numerical solut ion with

high-order schemesAt each iteration, at each cell, a new value for unknown variable in

cell P can then be calculated from equation:

U is so-called underrelaxation parameter. It is usually set form 0 to 1

Underrelaxation factors that are too small will significantly slow down

convergence, sometimes to the extent that the user thinks the solution is

converged when it really is not.

The recommendation is to always use underrelaxation factors that are ashigh as possible, without resulting in oscillations or divergence.

When the solution is converged but the pressure residual is still relatively

high, the factors for pressure and momentum can be lowered to further refine

the solution.

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The finite volume solution method can either use a segregated or a

coupled solution procedure.

With segregated methods an equation for a certain variable is solved for all

cells, then the equation for the next variable is solved for all cells, etc.

With coupled methods, for a given cell equations for all variables are solved,

and that process is then repeated for all cells.

The segregated solution method is the default method in most commercial

finite volume codes. It is best suited for incompressible flows or compressible

flows at low Mach number.

Compressible flows at high Mach number, especially when they involve

shock waves, are best solved with the coupled solver.

Solution methods for algebraic system generated in FVM

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Unsteady solution procedure

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Boundary conditions

Dirichlet boundary conditions: value of unknown function is

specified, e.g. velocity is zero at boundary

Neumann boundary conditions: gradient of unknownfunction is specified, e.g. temperature gradient is zero

Mixed boundary condition: linear combination of Dirichlet

and Neumann is set

At a given boundary, different types of boundary conditions can

be used for different variables

Adiabat ic BC is Dirichlet or Neumann or mixed ?

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Outflow boundaries cannot beused:

With compressible flows.

With the pressure inlet boundary

condition (use velocity inlet instead)

because the combination does not uniquely

set a pressure gradient over the whole

domain. In unsteady flows with variable

density.

Do not use outflow boundaries where:

Flow enters domain or when backflow

occurs (in that case

use pressure b.c.).

Gradients in flow direction are

significant.

Conditions downstream of exit plane

impact flow in domain.

Restrict ion on outf low boundary conditions

Can converged simulation with backflow represent a real flow ?

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Pressure boundary conditions

Incompressible flow Compressible flow

Backflow can occur at pressure outletboundaries:

During solution process or as part of solution.

Backflow is assumed to be normal to the

boundary.

Convergence difficulties minimized by realisticvalues for backflow quantities.

Value specified for static pressure used as

total pressure wherever backflow occurs.

Pressure outlet must always be used when

model is set up with a pressure inlet.

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Symmetry boundaries

Used to reduce computational effort in problem. Flow field and geometry must be symmetric:

Zero normal velocity at symmetry plane.

Zero normal gradients of all variables at

symmetry plane.

Also used to model slip walls in viscous flow.

Periodic boundaries

Used when physical geometry of interestand expected flow pattern and the thermal

solution are of a periodically repeating

nature.

Is it possible to have periodic boundaries

while modeling a heat transfer problem ?

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Axis boundary conditions

Used at the centerline (y=0 in Fluent)

of a 2-D axisymmetric grid.

CFX requires 3D mesh even for

axisymmetric flow

Axisymmetric BC is Dirichlet or Neumann or mixed ?

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Modeling the pressure drop through

a car catalytic converter when air

enters the inlet at 25 m/s, and exits

the outlet at a static pressure of 1

atm. Assume that the catalytic

converter contains isothermal air at

a temperature of 600 K.

CFX tutor ial: Flow in a Catalyt ic Converter

Quiz questions 1) where did you set under relaxation factor in this simulation? 2)

Cfd Class April 13 2010

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